Boundary Purity and Affine Collapse: A Proof of the Abhyankar-Sathaye Conjecture

This paper delivers a self-contained proof of the Abhyankar–Sathaye conjecture in dimension three: over an algebraically closed field of characteristic zero, every surjective polynomial map f: A³ₖ -> A¹ₖ whose general fiber is A²ₖ is shown to be a coordinate polynomial. The argument develops a vector-field approach built on shell-adapted compactifications, a rank-two translation lattice of vertical locally nilpotent derivations, and a boundary-purity analysis at the bad fibers, culminating in an affine collapse criterion that forces global triviality. In this sense, the work not only resolves a longstanding problem in affine algebraic geometry, but also offers a coherent geometric framework for turning local boundary control into a global coordinate conclusion.